Bleaching correction for DNA measurements in highly diluted solutions using confocal microscopy

Determining the concentration of nucleic acids in biological samples precisely and reliably still is a challenge. In particular when only very small sample quantities are available for analysis, the established fluorescence-based methods give insufficient results. Photobleaching is seen as the main reason for this. In this paper we present a method to correct for the photobleaching effect. Using confocal microscopy with single molecule sensitivity, we derived calibration curves from DNA solutions with defined fragment length. We analyzed dilution series over a wide range of concentrations (1 pg/μl—1000 pg/μl) and measured their specific diffusion coefficients employing fluorescence correlation spectroscopy. Using this information, we corrected the measured fluorescence intensity of the calibration solutions for photobleaching effects. We evaluated our method by analyzing a series of DNA mixtures of varying composition. For fragments smaller than 1000 bp, our method allows to determine sample concentrations with high precision in very small sample quantities (< 2 μl with concentrations < 20 pg/μl). Once the technical parameters are determined and remain stable in an established process, our improved calibration method will make measuring molecular biological samples of unknown sequence composition more efficient, accurate and sample-saving than previous methods.

In molecular biology, precise knowledge of molecule concentrations, especially nucleic 2 acid concentration, is required. Various molecular biological methods such as molecular 3 cloning or sequencing involve nucleic acids and depend on precise concentration 4 information [1,2]. Also, the analysis of tissue samples or the examination of expression 5 patterns of cells require exact concentration information on the extracted nucleic acid 6 amounts [3][4][5]. PCR based methods can analyze nucleic acid samples consisting of only 7 a few templates and are therefore widely used in molecular biology [5,6]. However, 8 these methods rely on sequence information from the sample. PCR based methods 9 cannot aid in the analysis of nucleic acid mixtures of unknown sequences. Fluorescence 10 measurements are highly promising for this task because of their extraordinary 11 sensitivity that even allows measurements of single molecule events [7]. A number of 12 fluorescent dyes is available that bind sequence independently to nucleic acids and thus 13 enable reliable labelling [8,9]. Additionally, fluorescent dyes for labeling are inexpensive 14 and easy to handle. To save as much of the valuable nucleic acid sample as possible, measurements in highly diluted solutions with microliter volumes are preferred [2]. 16 Confocal fluorescence microscopy can meet these challenges and is therefore the ideal 17 candidate for measuring mixtures of unknown nucleic acid sequences. 18 To obtain reliable results in fluorescence-based concentration determination, we have 19 to consider various effects. In confocal measurements soaring power intensities occur 20 which may be of the order of several 100 kW/cm 2 [10], which of course exposes the 21 fluorophores to an enormous load. This is why the phenomenon of photobleaching, 22 whereby molecules can permanently lose their fluorescence property due to irradiation 23 with excitation light [11], is particularly noteworthy. During confocal measurements on 24 freely diffusing molecules, a stationary equilibrium between the particle streams of 25 bleached and fluorescent molecules is established in the excitation volume, so that on 26 average an apparently lower intensity, and thus also a lower concentration, of 27 fluorophores in the solution is measured due to the bleaching effect.

28
The extent to which a measurement is influenced by photobleaching depends on 29 various factors, such as the irradiance of the excitation light, the photochemical 30 properties of the fluorophore used or the proportion of oxygen in the solvent [11]. The 31 probability that a fluorophore is bleached is directly related to the duration of 32 irradiation by the excitation light [12]. As a result, the bleaching rate occurring in the 33 solution is a function of the molecular size, since large molecules diffuse more slowly 34 than small molecules and thus remain longer in the excitation volume. The similar effect 35 is valid for mixtures of nucleic acids with arbitrary fragment length distribution. The 36 only difference is that the mean fragment length and the mean diffusion constant are the 37 characteristic properties of the mixture for photobleaching. Consequently, nucleic acid 38 mixtures of the same mass concentration but with different molarity have different mean 39 diffusion constants and due to photobleaching have different fluorescence intensities. 40 We have derived a new calibration procedure to account for this effect and to correct 41 the effect of photobleaching in fluorescence measurements. For this purpose, we use 42 fluorescence correlation spectroscopy in a confocal setup to analyze the diffusion 43 properties of DNA solutions. With this information, we can correct the fluorescence 44 measurements for photobleaching to achieve highly accurate results.

45
Fluorescence correlation spectroscopy (FCS) is a commonly used technique to analyze the diffusion behavior of particles. The idea behind FCS is the correlation in time of thermodynamic concentration fluctuations of diffusing particles in solution in equilibrium [13]. Eq 1 gives the autocorrelation function to calculate the FCS and corresponds to the correlation of a time series with itself shifted by time τ , as a function of τ : where I(t) is the measured intensity at time t. . . . t denotes a time average over time: The intensity fluctuation δI(t) at a certain time is: From this, the autocorrelation function turns into On basis of physical considerations, a model for the autocorrelation can be derived. For this, the properties of the laser profile and molecular diffusion properties of the sample are considered. This model gives ground to approximate the detection efficiency of a diffusing particle excited by a single-mode laser in a confocal setup by a Gaussian profile. [15]: where x, y and z are the coordinates of the observation volume. z is the direction of the laser beam. r 0 is the radius of the observation volume and (2 z 0 ) is the effective length of the volume. For the Gaussian distribution of the laser profile, we can write the autocorrelation function for one freely diffusing particle species as follows [16]: Here, N is the average molecule number in the detection volume. τ D is the average diffusion time of particles in the observation volume giving the characteristic decay scale of fluorescence fluctuations [14]. We use equation 6 to fit the experimental FCS data to get the diffusion time τ D from the measurement. At τ = 0, we obtain the mean number of diffusing particles in the volume N or equivalently the mean concentration C : Where the effective observation volume is: The diffusion coefficient of the diffusing molecules in solution is: Standard implementations of FCS methods do not provide absolute values of diffusion 47 coefficients, since they require information about the geometric shape of the detection 48 volume, which is challenging to measure independently. Therefore, measurements of the 49 diffusion coefficient are relative and require additional measurements of a reference 50 substance e.g. Alexa 488 with known diffusion coefficient and concentration to derive 51 the information about the geometric shape [14] .

52
It is important to note that with small shifting times further effects like e.g. triplet 53 state effects (microseconds) [17] or photodiode afterpulsing (nano-to microseconds) 54 [18] dominate the autocorrelation and Eq 6 needs to be adapted. Since in the present 55 work polymeric molecules with comparatively large diffusion times (τ D > 1 × 10 −3 s) 56 were analyzed, the important changes of the autocorrelation function take place at 57 relatively large shifting times and these effects acan be neglected.    [20] where D is the diffusion coefficient and M is the molecular 109 weigth. Actually, we found an exponent of −0.567 which is very close to previously 110 reported values of −0.57 [21] and −0.571 ± 0.014 [22] for dsDNA molecules in aqueous 111 solution (see figure 3). The exponent from the scaling law is independent of the ambient 112 temperature during the experiments. Therefore, we can directly compare the exponents. 113 The results of the diffusion measurements and their comparison to literature prove 114 the reliability of our measurements. We would like to point out that we have used the 115 diffusion time τ D instead of the diffusion coefficient D in the following sections. The resulting graph is almost linear and we can fit the data with a straight line with sufficient accuracy. At high concentrations, however, the graph is best described by a polynomial of 2nd order, since the quadratic term takes into account concentration-dependent effects such as quenching or volume exclusion. Therefore, we fit the dilution series of 50 bp DNA to a polynomial function the form I is the intensity and C is the concentration of the analysed solution. The y-axis intersections const is set to the backgound noise we observed during our measurements (16.417 kCps). Next, we rotate the fitted function I = f (C) around the z-axis intersection const to map the data of the other DNA-solutions of different fragment size. For this purpose we multiply the vector r = C I with a rotation matrix: where C and I are the measured concentration and intensity affected by photobleaching. By inserting I = f (C) in Eq 11 and by translating it into the origin, April 2, 2020 5/11 we obtain for the expressions C and I : Now, we want to express Eq 13 as a function of C . Therefore, we need to resolve Eq 120 12 to C: In our case θ lies between − arctan a < θ < π 2 − arctan a because physically only while b has to be minimal and negativ (|b| << a). Last but not least, only the negativ 125 term of Eq 14 is reasonable. Thus, we discart the positive term of Eq 14. Now, by 126 inserting Eq 14 into Eq 13 and by translating the expression back to const, we get the 127 following expression: 128 Eq 15 rotates Eq 10 around the z-axis intersection const. The validity of the approach is limited to functional areas where the rotated Eq 10 is monotonously growing. For larger concentrations the quadratic term starts to dominate and the approach is no longer valid. Eq 15 is subsequently fitted to the data of the remaining DNA dilution series (200 bp, 500 bp, 1000 bp, 2000 bp, 3000 bp, 6000 bp, 10 000 bp) to get the rotation angle θ for each fragment length (see Fig 5). We are aware of the fact that we can fit each dilution series directly to a polynomial function without the detour via rotation. But the procedure using one calibration curve and rotating it to fit the data seems to be much more stable. The resulting rotation angles θ for each dilution series provides the corresponding slope a sample . For this we rotate slope a around the angle θ.
a sample (θ) = sin θ + a cal cos θ cos θ − a cal sin θ = tan θ + a cal 1 − a cal tan θ Here, the slope a cal = a comes from the 50 bp dilution series in Fig 4. Now, we plot the resulting slope of each rotated curve against its diffusion time, respectively. The result can be seen in Fig 6. For modelling of the photobleaching as a function of τ D we employed a propability-based approach: Here N f is the average number of fluorescence photons and k f and k bl are the fluorescence rate and bleaching rate, respectively [23]. By setting k int = k f k bl and dividing the expression by τ D , we obtain the rate of fluorescent photons depending on the diffusion time. Finally, the introduction of a constant const is necessary to take into account the fact that the fluorescence rate for long diffusion times can never become zero but is approaching a limit value. Putting all these considerations together, Eq 17 turns into: The fit of Eq 18 to the data yields: k int = 0.1759, k bl = 19.0164 and const = 0.2571 129 (see Fig 6). The fluorescence intensity I is obtained directly from the measurements. The mean diffusion time τ D for a sample is then determined from the measurements using the autocorrelation. Eq 18 gives the characteristic slope a sample for a given diffusion time τ D . Now, resolving Eq 16 to the angle of rotation θ yields the reverse function θ: By inserting a sample from the step before and a cal from the 50 bp calibration 140 measurement, we get the resulting rotation angle θ of the sample. To calculate the 141 actual concentration of the DNA sample, we use the reverse function C(θ) of Eq 15: √ 2 sin θ × −8b const cot θ csc 5 θ + 8b I cot θ csc 5 θ + csc 6 θ + a 2 csc 6 θ − cos(2θ) csc 6 θ + a 2 cos(2θ) csc 6 θ + 2a csc 6 θ sin(2θ) April 2, 2020 7/11  concentration is significantly underestimated while the corrected calibration procedure 159 provides much more accurate results. 160 We believe that our improved calibration method will make measuring molecular 161 biological samples of unkown sequence composition effortless, accurate and 162 sample-saving when compared to previous methods. 163 We would like to point out that besides photobleaching, the so-called saturation of 164 optical intensity also plays an important role in fluorescence intensity. While the 165 saturation of the fluorescence intensity only occurs at soaring laser intensities, 166 significant parts of the molecules can already change into long-lasting triplet states at 167 consideralby lower laser powers. Since our measurements were all taken at the same 168 laser power, the relative deviation due to this effect is the same for all our 169 measurements and can be neglected for correction.

171
In this paper we derived a procedure to get highly accurate results for DNA 172 measurements in highly diluted solutions. The challenge is to correct the so called 173 photobleaching effect which reduces the fluorescence rate of the sample. The larger the 174 hydrodynamic radius of the sample the larger is the photobleaching effect. In a first 175 step, we determined the diffusion properties of the sample by means of fluorescence 176 correlation spectroscopy. By doing this we could correct the fluorescence rate of the 177 sample. In contrast to uncorrected measurements, we could reduce the photobleaching 178 dependent failure of fluorescence based measurements to less than 3 % compared to 179 30 % without correction. Even for very low sample concentrations of 20 pg/µl the failure 180 is still below 9 %. This is remarkable considering that we conducted the measurements 181 in tiny volumes of 2 µl. But even measurements in 1 µl droplet volumes are feasible.

182
Which means that in case of 20 pg/µl our method only needs 20 pg of DNA to provide 183 accurate results without the cost of expensive consumables. We would like to point out 184 that for each mixture the average diffusion time is measured which enables the calculation of the average DNA fragment size of the sample. In principle, this allows a 186 direct measurement of the sample molarity. It is also thinkable to determine the degree 187 of fragmentation of nucleic acids in a sample. This opens up interesting fields of 188 application in the field of DNA and RNA extraction from rare samples such as tissue 189 sections. Furthermore, this circumvents time-consuming and expensive examinations of 190 the sample using for example capillary electrophoretic methods.

Supporting information
192